Degree difference test

Statement

For rational functions

Consider a series of the form:

$\sum _{k=k_{0}}^{\infty }{\frac {p(k)}{q(k)}}$

where $p$ and $q$ are polynomials, $q$ is not the zero polynomial, and $q(k)$ is nonzero for all integers $k\geq k_{0}$. Then the following rules hold:

1. If $\deg q-\deg p\geq 2$, then the series is an absolutely convergent series.
2. If $\deg q-\deg p=1$, then the series diverges, but the terms limit to zero.
3. If $\deg p\geq degq$, then the series diverges, and the terms do not limit to zero.